The donkey has nothing on humans when it comes to being stubborn. During World War One a scientific test was undertaken to find whether donkeys could be used in combat. As it turned out donkeys are incredibly intelligent and manage to complete a series of puzzles and challenges including undoing a locked pack lock with their mouths. But as they are closely related to horses the tests were also carried out on horses and they were found to be even smarter and were used in many operations through out the war.
Marilyn Vos Savant
Perhaps you have heard of Marilyn vos Savant, national columnist and author, who was listed under "Highest IQ" for both childhood and adult scores, has been inducted into the "Guinness Hall of Fame" and was named in "Women of the New Millennium" by the White House Vital Voices: Women in Democracy campaign, as a winner of the "Women Making History" award from the National Women's History Museum and a recipient of Honorary Doctorates of Letters.
Since 1986, Marilyn has been writing the "Ask Marilyn" question-and-answer column for Parade, the Sunday magazine distributed by 379 newspapers, with a circulation of 34 million and a readership of 79 million, the largest periodical in the world. Questions from readers range from philosophical to mathematical to "just plain nuts," as Marilyn puts it. But, her real work takes place at Jarvik Heart, which manufactures artificial hearts for permanent and temporary use in the treatment of heart failure. Marilyn is married to Robert Jarvik MD, the inventor of the Jarvik 7 and Jarvik 2000 artificial hearts.
The Game Show Problem
In 1990, Marilyn answered a question from a reader who asked;
Suppose you’re on a game show, and you’re given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say #1, and the host, who knows what’s behind the doors, opens another door, say #3, which has a goat. He says to you, "Do you want to pick door #2?" Is it to your advantage to switch your choice of doors? ~ Craig F. Whitaker, Columbia, Maryland
Her answer was succinct;
Yes; you should switch. The first door has a 1/3 chance of winning, but the second door has a 2/3 chance. Here’s a good way to visualize what happened. Suppose there are a million doors, and you pick door #1. Then the host, who knows what’s behind the doors and will always avoid the one with the prize, opens them all except door #777,777. You’d switch to that door pretty fast, wouldn’t you?
This column proved to be one of Marilyn's most commented on bits of all time. At first it seemed that everyone thought she was wrong including the Deputy Director of the Center for Defense Information and a Research Mathematical Statistician from the National Institutes of Health! Of the letters from the general public, 92% were against Marilyn's answer, and of the letters from universities, 65% were against her answer. Overall, nine out of ten readers completely disagreed with Marilyn. Some of the letters were telling;
Since you seem to enjoy coming straight to the point, I’ll do the same. You blew it! Let me explain. If one door is shown to be a loser, that information changes the probability of either remaining choice, neither of which has any reason to be more likely, to 1/2. As a professional mathematician, I’m very concerned with the general public’s lack of mathematical skills. Please help by confessing your error and in the future being more careful. ~ Robert Sachs, Ph.D., George Mason University
You’re in error, but Albert Einstein earned a dearer place in the hearts of people after he admitted his errors. ~ Frank Rose, Ph.D., University of Michigan
I have been a faithful reader of your column, and I have not, until now, had any reason to doubt you. However, in this matter (for which I do have expertise), your answer is clearly at odds with the truth. ~ James Rauff, Ph.D., Millikin University
May I suggest that you obtain and refer to a standard textbook on probability before you try to answer a question of this type again? ~ Charles Reid, Ph.D., University of Florida
You made a mistake, but look at the positive side. If all those Ph.D.’s were wrong, the country would be in some very serious trouble. ~ Everett Harman, Ph.D., U.S. Army Research Institute
As it turns out Marilyn was, of course, correct in her reply, but the perceptions of thousands of extremely intelligent individuals kept them admitting that they could be wrong. After receiving so many letters of disagreement, Marilyn, explained in the next column.
My original answer is correct. But first, let me explain why your answer is wrong. The winning odds of 1/3 on the first choice can’t go up to 1/2 just because the host opens a losing door. To illustrate this, let’s say we play a shell game. You look away, and I put a pea under one of three shells. Then I ask you to put your finger on a shell. The odds that your choice contains a pea are 1/3, agreed? Then I simply lift up an empty shell from the remaining other two. As I can (and will) do this regardless of what you’ve chosen, we’ve learned nothing to allow us to revise the odds on the shell under your finger.
The benefits of switching are readily proven by playing through the six games that exhaust all the possibilities. For the first three games, you choose #1 and "switch" each time, for the second three games, you choose #1 and "stay" each time, and the host always opens a loser. Here are the results.
GAME DOOR 1 DOOR 2 DOOR 3 RESULT GAME 1 AUTO GOAT GOAT Switch and you lose. GAME 2 GOAT AUTO GOAT Switch and you win. GAME 3 GOAT GOAT AUTO Switch and you win. GAME 4 AUTO GOAT GOAT Stay and you win. GAME 5 GOAT AUTO GOAT Stay and you lose. GAME 6 GOAT GOAT AUTO Stay and you lose.
When you switch, you win 2/3 of the time and lose 1/3, but when you don’t switch, you only win 1/3 of the time and lose 2/3. You can try it yourself and see.
Alternatively, you can actually play the game with another person acting as the host with three playing cards—two jokers for the goat and an ace for the prize. However, doing this a few hundred times to get statistically valid results can get a little tedious, so perhaps you can assign it as extra credit—or for punishment! (That’ll get their goats!)
Still, the letters of disagreement flowed, forcing Marilyn to submit a second reply to her column;
Gasp! If this controversy continues, even the postman won’t be able to fit into the mailroom. I’m receiving thousands of letters, nearly all insisting that I’m wrong, including the Deputy Director of the Center for Defense Information and a Research Mathematical Statistician from the National Institutes of Health! Of the letters from the general public, 92% are against my answer, and and of the letters from universities, 65% are against my answer. Overall, nine out of ten readers completely disagree with my reply.
Now we’re receiving far more mail, and even newspaper columnists are joining in the fray! The day after the second column appeared, lights started flashing here at the magazine. Telephone calls poured into the switchboard, fax machines churned out copy, and the mailroom began to sink under its own weight. Incredulous at the response, we read wild accusations of intellectual irresponsibility, and, as the days went by, we were even more incredulous to read embarrassed retractions from some of those same people!
So let’s look at it again, remembering that the original answer defines certain conditions, the most significant of which is that the host always opens a losing door on purpose. (There’s no way he can always open a losing door by chance!) Anything else is a different question.
The original answer is still correct, and the key to it lies in the question, "Should you switch?" Suppose we pause at that point, and a UFO settles down onto the stage. A little green woman emerges, and the host asks her to point to one of the two unopened doors. The chances that she’ll randomly choose the one with the prize are 1/2, all right. But that’s because she lacks the advantage the original contestant had—the help of the host. (Try to forget any particular television show.)
When you first choose door #1 from three, there’s a 1/3 chance that the prize is behind that one and a 2/3 chance that it’s behind one of the others. But then the host steps in and gives you a clue. If the prize is behind #2, the host shows you #3, and if the prize is behind #3, the host shows you #2. So when you switch, you win if the prize is behind #2 or #3. You win either way! But if you don’t switch, you win only if the prize is behind door #1.
And as this problem is of such intense interest, I’m willing to put my thinking to the test with a nationwide experiment. This is a call to math classes all across the country. Set up a probability trial exactly as outlined below and send me a chart of all the games along with a cover letter repeating just how you did it so we can make sure the methods are consistent.
One student plays the contestant, and another, the host. Label three paper cups #1, #2, and #3. While the contestant looks away, the host randomly hides a penny under a cup by throwing a die until a 1, 2, or 3 comes up. Next, the contestant randomly points to a cup by throwing a die the same way. Then the host purposely lifts up a losing cup from the two unchosen. Lastly, the contestant "stays" and lifts up his original cup to see if it covers the penny. Play "not switching" two hundred times and keep track of how often the contestant wins.
Then test the other strategy. Play the game the same way until the last instruction, at which point the contestant instead "switches" and lifts up the cup not chosen by anyone to see if it covers the penny. Play "switching" two hundred times, also.
Then the unexpected happened. Educators, from middle school to the most prestigious universities begin performing experiments in the classroom. It is unclear whether they were performing the experiments to prove or disapprove Marilyn's statement, but in the end the tests were unanimous. Here is Marilyn's response;
Wow! What a response we received! It’s still coming in, but so many of you are so anxious to hear the results that we’ll stop tallying for a moment and take stock of the situation so far. We’ve received thousands of letters, and of the people who performed the experiment by hand as described, the results are close to unanimous: you win twice as often when you change doors. Nearly 100% of those readers now believe it pays to switch. (One is an eighth-grade math teacher who, despite data clearly supporting the position, simply refuses to believe it!)
But many people tried performing similar experiments on computers, fearlessly programming them in hundreds of different ways. Not surprisingly, they fared a little less well. Even so, about 97% of them now believe it pays to switch.
And plenty of people who didn’t perform the experiment wrote, too. Of the general public, about 56% now believe you should switch compared with only 8% before. And from academic institutions, about 71% now believe you should switch compared with only 35% before. (Many of them wrote to express utter amazement at the whole state of affairs, commenting that it altered their thinking dramatically, especially about the state of mathematical education in this country.) And a very small percentage of readers feel convinced that the furor is resulting from people not realizing that the host is opening a losing door on purpose. (But they haven’t read my mail! The great majority of people understand the conditions perfectly.)
And so we’ve made progress! Half of the readers whose letters were published in the previous columns have written to say they’ve changed their minds, and only one of them wrote to state that his position hadn’t changed at all.
A few of the letters from readers who actually attempted to disprove Marilyn wrote;
You are indeed correct. My colleagues at work had a ball with this problem, and I dare say that most of them, including me at first, thought you were wrong! - Seth Kalson, Ph.D., Massachusetts Institute of Technology
The teachers in my graduate-level mathematics classes, most of whom thought you were wrong, conducted your experiment as a class project. Each of the twenty-five teachers had students in their middle or high school classes play at least 400 games. In all, we had 14,800 samples of the experiment, and we’re convinced that you were correct —the contestant should switch! ~ Eloise Rudy, Furman University, Greenville, South Carolina
After considerable discussion and vacillation here at the Los Alamos National Laboratory, two of my colleagues independently programmed the problem, and in 1,000,000 trials, switching paid off. The total running time on the computer was less than one second. - G.P. DeVault, Ph.D., Los Alamos National Laboratory, Los Alamos, New Mexico
What does this say about the perceptions? Plenty, as there is a price to pay in not knowing the truth of things. Most of us hold the perception that not only do we not want to be wrong, we do not want the other person to be right, even when they are one of the most intelligent people on earth. And, that we will go to great extremes to prove that our perceptions are correct.
The fallacious and self-destructive ideas interfere with the ability to engage life effectively. But we are the sort of creatures that do not learn new things easily, if they contradict what we think we already know. We hear and read selectively. That is the way things are. It is so important for us to maintain our long-held perspective on ourselves and on the world, that we are more likely to martyr ourselves for our beliefs than admit to ourselves, and to others, that we may be wrong.
Perceptions can be altered. Anyone reading a newspaper can see reports of people, especially in groups, misperceiving events. Crowds look up at the sky and see stately space ships gliding by because everyone else in the crowd sees them also. The face of the Virgin Mary appears on a piece of toast and is venerated. Law schools have classes in which it is readily demonstrated that eye-witness testimony—however sincere—is often wrong. The growing number of convicts demonstrated by genetic evidence to be innocent of the crimes for which they have been convicted on eye-witness testimony demonstrates that fact over and over again. People claim to have been abducted by space aliens and interfered with sexually; and others believe them. Some individuals see ghosts, or communicate with the dead. And other people believe them. There are superstitions that hold many in thrall. I have patients (a number of them) who have to pick the “right clothes” to wear in the morning, lest something bad happen to someone in their families. And they truly believe that.
The world is full of people who believe opposite things. To mention a few: matters of religion, ethics, historical fact, politics, and even science, and so on. Our political scene is dominated by men and women who do not seem at first glance to differ much by education or intelligence coming down very strongly on opposite sides of the abortion and contraception controversy, the social obligations of government, the appropriateness of different military interventions at different times and, it seems, almost every other act of policy. None of us is surprised by these revelations as they come to our attention. They are what we have come to expect of each other. What is astonishing—at least to me—is how certain we are in our beliefs. Given the fact that we readily recognize folly in others, why can we not entertain the idea that we, ourselves, may be in error from time to time when we remember something, or when we think we experience something, or when we have come to believe something?
Believing is Seeing
What we believe is important because our experience is likely to be twisted in ways that support those beliefs. We perceive things we expect to perceive; and we then behave in ways consistent with those beliefs. These are sometimes called “self-fulfilling” predictions. What troubles people most often are precisely these preconceived notions, whether positive or negative.
Negative ideas are, of course, crippling. They undermine the confidence and lead to withdrawal and depression. They undermine any chance for success, whether it is success in business or in social relationships. Some will see psychotherapists in an attempt to better see himself/herself, and the world in general, more accurately. However, often the fact that these ideas are held to stubbornly, interferes with their therapy. “I know the way I am,” people sometimes say, to devastating effect. If they cannot be different, their experience of the world cannot change.
It would be nice also, I can’t help thinking, if everyone was less certain of being right all the time. We would all get along better. I am tired, though, of inveighing against the self-righteousness of others and the smug certainty of ignorant people. I have no reason to think I am any better than they are. In fact, I have some reasons to think I may be worse. I know when I am proven wrong about something, I am bothered more than other people would be.
Since these defects of thinking seem to be part of the human condition, it is worth wondering why. Why do some people hold onto their ideas in the face of overwhelming evidence of being wrong? The best examples of these are scientists, who, after all, are supposed to be dealing with objective fact. Still, those geologists who ridiculed the idea of continental drift never changed their minds. They died first. Similarly, the theory of relativity when it first was published was dismissed by some of the world’s great physicists. And some of them continued to feel the same way long after everyone else was convinced of its accuracy. Just as a body in motion tends to remain in motion, beliefs tend to continue even when contradicted by facts.
I think these are some of the reasons:
- We are pack animals, much like the donkey discussed at the beginning of the column. We have evolved to live and work in groups, and we are inclined to share ideas and ideologies intrinsic to those groups, including among them religious and political beliefs. There is a survival advantage to belonging. We feel comfortable with others who think the same way. Sometimes reality is shunted aside.
- Most of the time what we learn from others is reliable. Think of the weatherman or the doctor. Skepticism is less useful more of the time than the willingness to be instructed by people in whom we believe. Sometimes, obviously, we are led astray, but there is an advantage in holding to ideas that have formed over a long period of time, instead of being pulled in different directions all the time. It is anxiety-provoking to be in doubt.
- Many fixed ideas are self-congratulatory. We feel morally superior. We know the truth. That is a comforting idea and colors our reaction to anything that seems contrary to what we believe. (Think of the cable news channels. Conservatives tend to watch the same news channel all the time while liberals watch a different one.) It is embarrassing to be wrong about important matters, especially when we have expressed ourselves strongly about them.
- Changing our attitudes and ideas is likely to imply that we should change our behavior. Behaving differently risks failure and ridicule. It is easier to justify to ourselves continuing to do whatever we are accustomed to doing. We have a stake in being right.
- Perception is so malleable, we miss the chance to learn something new at the very first instant of hearing it or seeing it. Seeing is not believing. It is the other way around. Believing is seeing. All of experience tends to confirm what we already think.
There is a price to pay in not knowing the truth of things. The fallacious and self-destructive ideas that most psychotherapy patients have, some of which I mention above, are good examples. They interfere with the ability to engage life effectively. But we are the sort of creatures that do not learn new things easily, if they contradict what we think we already know. We hear and read selectively. That is the way things are. It is so important for us to maintain our long-held perspective on ourselves and on the world, that we are more likely to martyr ourselves for our beliefs than admit to ourselves, and to others, that we may be wrong.
Parade Magazine, 1990-1991, Marilyn vos Savant
Fighting Fear, Fredric Neuman, M.D.